Questions tagged [steins-phenomenon]

Stein's phenomenon (paradox) states that when three or more parameters are estimated at the same time, there are more accurate estimators than the average over all observations.

Filter by
Sorted by
Tagged with
1
vote
0answers
28 views

Is the shrinkage of subgroup analyses in meta-analysis an example of Stein's paradox?

This paper writes (edited for concision): Consider, a doctor in Germany confronted by a meta-analysis of long term‚ $\beta$ blockade after myocardial infarction. Although a robust beneficial effect ...
2
votes
0answers
80 views

James Stein Estimator for more than one Sample

I have a hard time understanding the James-Stein Estimator. I show you how I tried to comprehend it by using a python example. I take a normally distributed random vector with mean $(0.1, 0.2, 0.3, 0....
1
vote
1answer
79 views

Dominating Positive Part James-Stein

Is dominating Positive Part James Stein estimator when estimating the mean of a multivariate normal of dimension 3 with known variance(all equal) an open problem? If not, what is this estimator ...
1
vote
0answers
19 views

Clarifying a proof of a particular paper on Steins Estimator

I am trying proving result (5.4) of the following paper. Its a paper on Steins estimator on spherically symmetric cases. The doubt is a s follows: Given $$X|\theta\sim \mathcal{N}(\theta,I)$$ ...
6
votes
1answer
213 views

James-Stein Estimator with unequal variances (Ch. 2)

After studying James-Stein estimators for a few weeks and looking at many different sources I am stuck at trying to understand how Efron and Morris calculated the Toxoplasmosis example in their 1975 ...
19
votes
1answer
6k views

Why is the James-Stein estimator called a “shrinkage” estimator?

I have been reading about the James-Stein estimator. It is defined, in this notes, as $$ \hat{\theta}=\left(1 - \frac{p-2}{\|X\|^2}\right)X$$ I have read the proof but I don't understand the ...
4
votes
0answers
58 views

When does $\forall p: E_p[f] = E_p[g]$ imply $f = g$?

Let's say $$E_p[f(X)] = E_p[g(X)]$$ for all $p \in S$, where $S$ might be some parametric family of densities, for instance. Under which assumptions on $S$ does this imply $f = g$? I am reading Efron ...
2
votes
1answer
93 views

James-Stein Estimator with unequal numbers in groups

In the book Computer Age Statistical Inference the James-Stein estimator is introduced. Brad Efron runs through an example where batting averages are estimated from each players 90 at-bats. $$p_i\sim ...
5
votes
1answer
202 views

Stein's estimator normality assumption

The Stein's estimator assumes that data points are draws from a normal distribution, i.e., $Z_i \sim N(\mu_i, \sigma^2_i)$. By looking at different sources (Wikipedia, Efron,James-Stein Estimator ...
1
vote
0answers
50 views

Estimator of a normal mean: practical recommendation?

I have a sample of $n=370$ $5$-dimensional observations which can be reasonably well modeled using a multivariate normal distribution. I want to estimate the mean using this sample, which will be part ...
7
votes
1answer
504 views

Shrinkage of the eigenvalues

Assume we have $n$ samples $X_1,..., X_n$ which are independent and identically distributed with mean = 0 and unknown non-singular covariance matrix $M$. Each sample $X_i$ is a vector of size $p\times ...
4
votes
1answer
469 views

Stein's estimator vs James-Stein estimator

I read a lot of sources concerning stein's estimator and James-Stein estimator. Unfortunately, a lot of sources do not write the correct formulas of each estimator. And so I am now confused!! Kindly, ...
4
votes
1answer
242 views

Shrinkage estimation of Efron and Morris (1972)

I read this article: Article1 and which was refined by the second article Article2 that was considered as a generalization of the James-Stein estimator. In article 1 for example, they considered the ...
3
votes
2answers
1k views

Admissible Estimator for Linear Regression

Is there an admissible estimator for a linear regression model with many parameters without restricting the parameter space? Admissibility will be with respect to Mean Square Error on the regression ...
3
votes
0answers
125 views

Stein's Paradox and credibility theory

Stein's paradox says that The ordinary decision rule for estimating the mean of a multivariate Gaussian distribution is inadmissible under mean squared error risk in dimension at least 3 (...
20
votes
1answer
374 views

Does Stein's Paradox still hold when using the $l_1$ norm instead of the $l_2$ norm?

Stein's Paradox shows that when three or more parameters are estimated simultaneously, there exist combined estimators more accurate on average (that is, having lower expected mean squared error) than ...
3
votes
0answers
85 views

What is the relationship, if any, between Stein's Paradox and linear restrictions in regressions?

Suppose we have $$y = b_1x_1 + b_2x_2 + b_3x_3 + e$$ as our regression model. Setting a linear restriction, say $b_1 + b_2 + b_3 = 0$, allow us to rewrite the model as, $$y = (b_1)(x_1 - x_3) + (...
65
votes
5answers
4k views

Unified view on shrinkage: what is the relation (if any) between Stein's paradox, ridge regression, and random effects in mixed models?

Consider the following three phenomena. Stein's paradox: given some data from multivariate normal distribution in $\mathbb R^n, \: n\ge 3$, sample mean is not a very good estimator of the true mean. ...
11
votes
1answer
2k views

James-Stein Estimator with unequal variances

Every statement I find of the James-Stein estimator assumes that the random variables being estimated have the same (and unit) variance. But all of these examples also mention that the JS estimator ...
5
votes
1answer
355 views

When estimating population mean, how can one half of the sample mean have lower risk than the sample mean itself?

I read Efron and Morris (1977) Stein's Paradox in Statistics with interest yesterday and stumbled upon the statement that, if and only if the population mean is close to zero, than the risk (mean ...
12
votes
1answer
2k views

Is there a connection between empirical Bayes and random effects?

I recently happened to read about empirical Bayes (Casella, 1985, An introduction to empirical Bayes data analysis) and it looked a lot like random effects model; in that both have estimates shrunken ...
1
vote
1answer
235 views

Are there any extensions of the James-Stein estimator for the case of dependent variables?

Are there any extensions of the James-Stein estimator for the case of dependent variables? I've done numeric experiments with the James-Stein estimator on correlated normal variables (5-10 variables,...
47
votes
2answers
4k views

Intuition behind why Stein's paradox only applies in dimensions $\ge 3$

Stein's Example shows that the maximum likelihood estimate of $n$ normally distributed variables with means $\mu_1,\ldots,\mu_n$ and variances $1$ is inadmissible (under a square loss function) iff $n\...
18
votes
2answers
5k views

James-Stein estimator: How did Efron and Morris calculate $\sigma^2$ in shrinkage factor for their baseball example?

I have a question on calculating James-Stein Shrinkage factor in the 1977 Scientific American paper by Bradley Efron and Carl Morris, "Stein's Paradox in Statistics". I gathered the data for the ...
15
votes
5answers
2k views

James-Stein shrinkage 'in the wild'?

I am taken by the idea of James-Stein shrinkage (i.e. that a nonlinear function of a single observation of a vector of possibly independent normals can be a better estimator of the means of the random ...